"How do I show that a two-qubit state is an entangled state?" includes an answer which references the Peres–Horodecki criterion. This works for $2\times 2$ and $2\times3$ dimensional cases; however, in higher dimensions, it is "inconclusive." It is suggested to supplement with more advanced tests, such as those based on entanglement witness. How would this be done? Are there alternative ways to go about this?
Determining whether a given state is entangled or not is NP hard. So if you include all possible types on entanglement, including mixed states and multipartite entanglement, there is never going to be an elegant solution. Techniques are therefore defined for specific cases, where the structure of the problem can be used to create an efficient solution.
For example, if a state is bipartite and pure, you can simply take the reduced density matrix of one party and see if it is mixed. This could be done by computing the Von Neumann entropy to see if it is non-zero (this quantity provides a measure of entanglement in this case).
This approach would work for any pure state of two particles, whatever their dimension. It can also be used to calculate entanglement for any bipartition. For example, if you had $n$ particles, you could take the first $m$ to be one party, and the remaining $n-m$ to be another, and use this technique to see if any entanglement exists between these groups.
For other cases, the approach you take will depend on the kind of entanglement you are looking for.
As suggested in your Wiki link, the way to detect an entangled state is to find a hyperplane that separates it from the convex set of separable states. This hyperplane represents what is called an entanglement witness. The PPT criterion that you mentioned is one such witness. Now to construct entanglement witnesses for higher dimensional systems is not easy, but it can be done algorithmically by solving a hierarchy semi-definite programs (SDP) . This hierarchy is complete, as every entangled state will eventually be detected. But it is computationally inefficient if the entangled state is very close to the convex set of separable states. It is infact known that detecting entanglement is NP-hard.
Doherty, Andrew C., Pablo A. Parrilo, and Federico M. Spedalieri. "Complete family of separability criteria." Physical Review A69.2 (2004): 022308
Gharibian, Sevag. "Strong NP-hardness of the quantum separability problem." arXiv preprint arXiv:0810.4507 (2008).